Data about hyperbolic Coxeter systems

TL;DR

This paper compiles detailed data on hyperbolic Coxeter systems, explicitly computing their Poincaré series and growth rates using Magma, aiding in understanding their algebraic and geometric properties.

Contribution

It provides the first explicit computations of Poincaré series and growth rates for minimal hyperbolic Coxeter systems, facilitating further algebraic and geometric analysis.

Findings

Explicit formulas for Poincaré series of hyperbolic Coxeter systems

Computed growth rates for these systems

Initial coefficients of the series are provided for recurrence relations

Abstract

We collect several data about Coxeter systems (cf. [Bou07, Hum90]), with particular emphasis on the hyperbolic ones. For each ($\preceq$-minimal) hyperbolic Coxeter system (W,S) the Poincar\'e series \[p_{(W,S)}(t)=\sum_{w\in W} t^{\ell(w)}\] and the growth rate \[ \omega(W,S)=\limsup_n \sqrt[n]{a_n}\] are explicitly computed using Magma (cf. [BCP97]). These computations were performed in connection to the proof of [Ter, Thm. B]. Since the Poincar\'e series represents a rational function, one may recover the sequence $(a_k)_{k\geq 0}$ through a linear recurrence relation on the coefficients, provided that enough terms at the beginning of the sequence are known. For each Coxeter system the initial coefficients $(a_k)_{k=0}^N$ are computed, where $N$ is the degree of the numerator of $p_{(W,S)}(t)$.